Project 1: X-Ray Diffraction from orderered and disordered 1D lattices

Due to the fact that perfect crystals are periodic, Fourier transform techniques are one of the basic tool to study such systems. Experimentalist and theorists often use Fast Fourier Transform packages, without worrying about details of their inner working, to perform spectral analysis and study dynamical systems. In the context of crystals, these techniques illustrate the basic ideas of reciprocal space and diffraction patterns in a particularly transparent way.

(a) Perfect Crystals: The diffraction patter of perfect crystal is a set of $\delta$-function peaks at the reciprocal lattice points. Consider a one-dimensional crystal with atoms centered at $R_n$ whose electron density form Gaussians of width $\sigma_n$

\begin{displaymath} \rho(x) = \sum_n e^{-(x-R_n)^2/2\sigma_n^2} \end{displaymath}

 

Assume that lattice constant is $a=2.3$ and $\sigma_n=0.2$. Start with a vector of values of $\rho$ containing 256 entries, which span 32 lattice periods. This vector should also be periodic, i.e., $\rho(0)=\rho(256)$. The sum over $n$ need not extend far beyond the range of $x$. Calculate the Fourier transform $\rho(k)$ using the FFT routine. Confirm that electron density $\rho(x)$ looks like an array of atoms (around each atom you should see a Gaussian distribution of charge density). Plot the power spectrum $\vert\rho(k)\vert^2$ over a range which includes the first four Bragg peaks.

(b) Diatomic Basis: If the unit cell of a crystal doubles in size, the reciprocal lattice divides in two. Let $\sigma_n$ be 0.1 if $n$ is even, and $\sigma_n=0.3$ if $n$ is odd. Examine your real-space density; for the same range of $x$ (now 16 lattice periods) and the same parameters, plot the power spectrum.

(c) Thermal Motion: At finite temperatures, the atoms will oscillate about their lattice positions: does the diffraction pattern get destroyed? Add a random number $\epsilon$ with standard deviation $\sigma_\epsilon$ to the positions, $R_n = n a + \sigma_\epsilon {\rm rand}$, where random numbers are chose from a normal probability distribution

\begin{displaymath} {\rm Probability(rand)} = \frac{e^{-{\rm rand}^2/2}}{\sqrt{2\pi}} \end{displaymath} (2)

Keeping $\sigma_x=0.2$, examine the real-space densities and plot the power spectrum for $\sigma_\epsilon=0.1$ and $\sigma_\epsilon=0.5$. Show analytically the the reciprocal lattice peaks stay sharp, but that they are decreased in magnitude by $e^{-\sigma_\epsilon^2 k^2}$: start from the definition


\begin{displaymath} \rho(k) = \int dx \, e^{ikx} \sum_n \rho_0 (x-na-\epsilon_n) \end{displaymath} (3)

pull out the terms involving $\epsilon_n$, and change the sum over n into an integral over $\epsilon$ involving ${\rm Probability}(\epsilon)$ (note that the final thing to calculate is the power spectrum is $\vert\rho(k)\vert^2$). Compare your numerical results to the analytical prediction (the so-called Debye-Weller factor).

(d) Glasses: A more realistic model of thermal motions would have the positions of the atoms vary with respect to their neighbors, $R_n - R_{n-1} = a + \sigma_\epsilon {\rm rand}$, rather than with respect to an invisible ideal lattice. Plot the power spectrum for $\sigma_\epsilon=0.05$ and $\sigma_\epsilon=0.2$. In one and two dimensions crystalline long-range translational order does not exist for $T>0$. In three dimensions, an atom has more neighbors far away who remember where it is supposed to sit, and a reasonably high temperature is needed to destroy all remnants of periodicity. The model in part (c) is thus reasonable for 3D crystals; it is Einstein's model for phonons in diamond. The model in part (d) gives a pretty good description of glasses and liquids: the Bragg peaks become broad when the long-range order disappears.

NOTE: The final solution for each system listed below should be presented in the form of a relevant plot (i.e., $\rho(x)$ vs. $x$ and $\vert\rho(k)\vert^2$ vs. $k$). The FFT routines and uniform random number generators are available in Maple, Mathematica, and on the Course Web site (in Fortran or C, from Numerical Recipes). The Project report should emulate the format of a research paper (with abstract, PACS codes, introduction, results, and conclusion - for an example of such format see Student Report).

HINT: Problems in Chapter 1 of L. L. Mihály and M. Martin: Solid State Physics Problems and Solutions.

 

 

 

 

 

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